Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/111

Line-Integral of Electric Force, or Electromotive Force along an Arc of a Curve.

69.] The Electromotive force along a given arc $AP$ of a curve is numerically measured by the work which would be done on a unit of positive electricity carried along the curve from the beginning, $A$, to $P$, the end of the arc.

If $s$ is the length of the arc, measured from $A$, and if the resultant force $R$ at any point of the curve makes an angle $c$ with the tangent drawn in the positive direction, then the work done on unit of electricity in moving along the element of the curve ${ds}$ will be

$R\cos \epsilon \,ds$

,

and the total electromotive force $V$ will be

$V=\int _{0}^{s}R\cos \epsilon \,ds$,

the integration being extended from the beginning to the end of the arc.

If we make use of the components of the force $R$, we find

$V\int _{0}^{s}(X{\frac {dx}{dy}}+Y{\frac {dy}{ds}}+Z{\frac {dz}{ds}})\,ds.$.

If $X,Y,$ and $Z$ are such that $X{dx}+Y{dy}+Z{dz}$ is a complete differential of a function of$x,y,z,$ then

$V=\int _{A}^{P}(X\,dx+Y\,dy+Z\,dz)=V_{A}-V_{P}$;

where the integration is performed in any way from the point $A$ to the point $P$, whether along the given curve or along any other line between $A$ and $P$.

In this case $V$ is a scalar function of the position of a point in space, that is, when we know the coordinates of the point, the value of $V$ is determinate, and this value is independent of the position and direction of the axes of reference. See Art. 16.

On Functions of the Position of a Point.

In what follows, when we describe a quantity as a function of the position of a point, we mean that for every position of the point the function has a determinate value. We do not imply that this value can always be expressed by the same formula for all points of space, for it may be expressed by one formula on one side of a given surface and by another formula on the other side.